3.3 Proofs for Section 2.3 “Linear Time-Variant Filtering of GACS Processes”
3.3.1 Proof of (2.112)
By substituting (2.111) into (2.110) we have
where, in the second equality the variable changes s1 = s + u, s2 = u are made. Furthermore, by making the variable change 
into the inner integral in (3.33), we have
where 
is defined in (2.113).
By substituting (3.34) into (3.33), we obtain (2.112).
Assumption (1.47) allows to interchange the order of integral and expectation operators (Fubini and Tonelli Theorem (Champeney 1990, Chapter 3)).
3.3.2 Proof of (2.117)
From (2.112) with τ1 = τ and τ2 = 0 it follows that
from which (2.117) immediately follows.
To obtain (3.35), under assumptions (2.114)–(2.116), the order of
operations can be interchanged. In fact, the following inequalities hold
(3.36)
and
(3.37)
independent of T.
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